MHB Maximum Likelihood Estimators for Uniform Distribution

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To find the maximum likelihood estimators for a sample of size n from a uniform distribution U(0, θ], the likelihood function is given by L(θ) = 1/θ^n. The logarithm of the likelihood function is ln(L(θ)) = -n ln(θ), and its derivative with respect to θ leads to the equation -n/θ = 0, which suggests an issue with estimating θ directly. The key point is that θ must be at least as large as the maximum value in the sample, as any sample value greater than θ would result in a likelihood of zero. Therefore, the maximum likelihood estimator for θ is the maximum observed value in the sample.
Julio1
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Find maximum likelihood estimators of an sample of size $n$ if $X\sim U(0,\theta].$

Hello MHB :)! Can any user help me please :)! I don't how follow...
 
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Julio said:
Find maximum likelihood estimators of an sample of size $n$ if $X\sim U(0,\theta].$

Hello MHB :)! Can any user help me please :)! I don't how follow...

Hi Julio!

We want to maximize the likelihood that some $\theta$ is the right one.

The likelihood that a certain $\theta$ is the right one given a random sample is:
$$\mathcal L(\theta; x_1, ..., x_n) = f(x_1|\theta) \times f(x_2|\theta) \times ... \times f(x_n|\theta)$$
where $f$ is the probability density function.

Since $X\sim U(0,\theta]$, $f$ is given by:
$$f(x|\theta)=\begin{cases}\frac 1 \theta&\text{if }0 < x \le \theta \\ 0 &\text{otherwise}\end{cases}$$

Can you tell for which $\theta$ the likelihood will be at its maximum?
 
Thanks I like Serena :).

Good have that the likelihood function is $L(\theta)=\dfrac{1}{\theta^n}.$ Then applying logarithm we have that

$\ln (L(\theta))=\ln(\dfrac{1}{\theta^n})=-n\ln(\theta).$ Now for derivation with respect $\theta$ we have that $\dfrac{\partial}{\partial \theta}(\ln L(\theta))=\dfrac{\partial}{\partial \theta}(-n\ln(\theta))=-\dfrac{n}{\theta}.$ Thus, match with zero have $-\dfrac{n}{\theta}=0$, i.e., $n=0$.

But why? :(, remove the parameter $\theta$?... Then I don't find an estimator for $\theta$?
 
You're welcome Julio!

What's missing in your approach is that it's not taken into account that the function is piecewise.
So we need to inspect what happens at the boundaires.

Note that any $x_i$ that is in the sample has to be $\le \theta$, because otherwise its probability is $0$.
So $\theta$ has to be at least the maximum value that is in the sample.
What happens to the likelihood if $\theta$ is bigger than that maximum value?
 
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